```python
from typing import Optional
from pydantic import BaseModel
from datetime import datetime

class PostBase(BaseModel):
    title: str
    content: str

class PostCreate(PostBase):
    pass

class PostUpdate(BaseModel):
    title: Optional[str] = None
    content: Optional[str] = None

class PostInDBBase(PostBase):
    id: int
    created_at: datetime
    updated_at: Optional[datetime] = None

    class Config:
        orm_mode = True

class Post(PostInDBBase):
    pass

class PostInDB(PostInDBBase):
    pass
```

Explanation:

1. We import the necessary modules: `typing` for type hints, `pydantic` for defining data models, and `datetime` for working with date and time.


3. `PostCreate` inherits from `PostBase` and is used for creating a new post.


5. `PostInDBBase` inherits from `PostBase` and adds fields specific to the database representation: `id`, `created_at`, and `updated_at`. The `Config` class is used to enable the ORM mode, which allows Pydantic to work with SQLAlchemy models.

6. `Post` and `PostInDB` are response schemas that inherit from `PostInDBBase`. These schemas will be used to serialize the data returned from the API.